When you act, are you a character or just a monster? What I mean is, do you actually give your monster a personality, or do you just pop out and scream? Is a personality really needed in a haunted house?

what you are asking is "Is the color blue necessary?" A character may not be necessary for some actors and integral to the reason why others haunt in the first place. Customers are the same way- it may be enough for some that you look scary and move right, others might only get scared when they see relationships between actors and learn that "junior" wants to have quality time with a velour robe, a Barry White album, and their corpse. Characters add depth to a haunts atmosphere.
Personally when I work a haunt I do both. i work a section of the haunt and am several characters or monsters depending on how I interact with the guests. If they see my full body then i am that areas character, if it is just my hand through a window I will make a zombie sound as I reach through, if i am just flipping a curtain at them then i might just make animalistic noises at them, If I am in a pitch black hall with them then i may use a different voice and say scary things. It keeps it fresh for me and the customer thinks they interacted with more than one actor.
Allen H